TM1: Set Theory + Binary Relations

When: Week 7 (after Test 2) Duration: 120 minutes Format: Closed book

Coverage

Set Theory (Weeks 1-2, 6)

Set operations and laws
Power sets
Cartesian products
Russell’s paradox
Axiomatic set theory (ZFC)
Cardinality
Cantor’s arguments
Schroeder-Bernstein theorem

Binary Relations (Weeks 3-7)

Properties of relations
Equivalence relations ↔ partitions
Order relations
Hasse diagrams
Functions and their properties
Composition and inverses
Lattices and Boolean algebras

Sample Questions

Definitions

Define equivalence relation. Give example.
What is a well-ordering?
Define power set. What is |𝒫(A)| if |A| = n?

Theorems

State and prove: equivalence ↔ partition
State Cantor’s theorem. Sketch proof.
Prove composition of functions is associative.

Proofs

Prove ℝ is uncountable.
Show: if f, g bijective, then g ∘ f bijective.
Prove every finite poset has maximal element.

Conceptual

Explain why Russell’s paradox matters.
Difference between maximal and greatest element?
Example: injective but not surjective function.

Preparation

Review all definitions (can recite precisely?)
Know major theorem statements
Practice proof techniques
Create concept maps
Form study groups
Attend review session